Momentum Eigenstates

Being in a momentum eigenstate in quantum mechanics means that the wave function of the particle is a plane wave of the form $\psi(x) = e^{ikx}$, where $k$ is the wave number related to the momentum of the particle by $p = \hbar k$. This form is a solution to the time-independent Schrödinger equation for a free particle, where the potential energy $V(x)$ is zero.

However, plane waves such as $e^{ikx}$ are not square integrable over all space, which means they do not belong to the space of $L^2(\mathbb{R})$ functions. Physically, this implies that a particle in a pure momentum eigenstate cannot be localized in space; the probability of finding the particle at any specific location is constant everywhere.

Despite not being normalizable, momentum eigenstates are still useful. They form a basis for the space of square integrable functions due to the completeness of the set of plane waves. This means any physical wave function $\psi(x)$ that describes a quantum state can be expressed as a superposition (integral) of these plane waves, known as a Fourier transform. This superposition, or wave packet, is square integrable and localizable, making it a more physically realistic state of the particle. 

The wave packet itself is not an eigenstate of the momentum operator $\hat{p} = -i\hbar \frac{\partial}{\partial x}$ since it is a combination of multiple momentum eigenstates with different $k$ values. Consequently, a wave packet has a spread in momentum and, due to the Heisenberg Uncertainty Principle, also a spread in position, which allows for the particle to be localized to a region in space.

The spectrum of the momentum operator in this context is continuous, which means that the eigenvalues $k$ can take any real value, leading to the continuous nature of possible momentum values for a quantum state in free space.

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